It was Monaco's second win in three league games under coach Thierry Henry and helped the club leapfrog Amiens and Dijon and move into 17th place in the 20-team table.
The 2017 league champions went ahead with a Falcao penalty two minutes before halftime and the Colombian striker then doubled their lead six minutes into second-half stoppage time when he got a second goal from the spot.
Amiens, who have now lost six of their last seven league games, were the better team throughout but profligate finishing and defensive errors cost them dearly.
Their miserable night was completed when Bakaye Dibassy was shown a red card seconds from the final whistle.
Elsewhere in Ligue 1, Nice drew 0-0 with Angers, while second-placed Montpellier are at home to fourth-placed Lille in the later match.
Runaway leaders Paris St Germain play Strasbourg on Wednesday.